A directed graph G has 5 vertices, numbered 1 through 5. The 5 x 5 matrix A is the adjacency matrix for G. The matrices A2 and A3 are given below. 0 0 1 0 0 0 0 0 1 0 A² = 0 0 0 0 1 A3= 10100 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 10100 0 1 0 1 0 0 0 1 0 1 Which edge is not in the transitive closure of G? (2,4) ○ (5,5) ○ (1,3) ○ (4,5)

Let H be a graph with the following adjacency matrix: 0 1 1 1] 01 1001 [1 1 1 (a) Draw H. Is H a tree? Justify your answer. A = (b) Compute A². What is the meaning of the number in the first row and first column of A²? (c) Could the matrix A² be the adjacency matrix of a simple graph? If so, draw the graph. If not, explain why.

Find, if possible, an invertible matrix P and a diago- nal matrix D such that A = PDP-. Otherwise, explain why A is not diagonalizable. 5 1 2 1 4 1 -3 -2 0 -(t - 3)³

Consider the adjacency matrix 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 of a graph with nodes labeled a,b,c,d,e and f and lexiographically ordered rows and columns. 1 0 1 1 0 0 Lo 1 0 1 0 0 Suppose the above adjacency matrix identifies the network in a small company, where nodes represent workers and edges indicate that those workers interact during the week. The company has a single team leader and two different projects, one larger than the other. Workers in the same project interact often during the week. b) State whether all workers are involved in at least one project and whether there is any worker, apart from the team leader, who works simultaneously on both projects.